Table 1

Standard Normal Distribution Table ϕ(z)=P [X≤ z]

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.50000 0.50399 0.50798 0.51197 0.51595 0.51994 0.52392 0.52790 0.53188 0.53586
0.1 0.53983 0.54380 0.54776 0.55172 0.55567 0.55962 0.56356 0.56749 0.57142 0.57535
0.2 0.57926 0.58317 0.58706 0.59095 0.59483 0.59871 0.60257 0.60642 0.61026 0.61409
0.3 0.61791 0.62172 0.62552 0.62930 0.63307 0.63683 0.64058 0.64431 0.64803 0.65173
0.4 0.65542 0.65910 0.66276 0.66640 0.67003 0.67364 0.67724 0.68082 0.68439 0.68793
0.5 0.69146 0.69497 0.69847 0.70194 0.70540 0.70884 0.71226 0.71566 0.71904 0.72240
0.6 0.72575 0.72907 0.73237 0.73565 0.73891 0.74215 0.74537 0.74857 0.75175 0.75490
0.7 0.75804 0.76115 0.76424 0.76730 0.77035 0.77337 0.77637 0.77935 0.78230 0.78524
0.8 0.78814 0.79103 0.79389 0.79673 0.79955 0.80234 0.80511 0.80785 0.81057 0.81327
0.9 0.81594 0.81859 0.82121 0.82381 0.82639 0.82894 0.83147 0.83398 0.83646 0.83891
1.0 0.84134 0.84375 0.84614 0.84849 0.85083 0.85314 0.85543 0.85769 0.85993 0.86214
1.1 0.86433 0.86650 0.86864 0.87076 0.87286 0.87493 0.87698 0.87900 0.88100 0.88298
1.2 0.88493 0.88686 0.88877 0.89065 0.89251 0.89435 0.89617 0.89796 0.89973 0.90147
1.3 0.90320 0.90490 0.90658 0.90824 0.90988 0.91149 0.91309 0.91466 0.91621 0.91774
1.4 0.91924 0.92073 0.92220 0.92364 0.92507 0.92647 0.92785 0.92922 0.93056 0.93189
1.5 0.93319 0.93448 0.93574 0.93699 0.93822 0.93943 0.94062 0.94179 0.94295 0.94408
1.6 0.94520 0.94630 0.94738 0.94845 0.94950 0.95053 0.95154 0.95254 0.95352 0.95449
1.7 0.95543 0.95637 0.95728 0.95818 0.95907 0.95994 0.96080 0.96164 0.96246 0.96327
1.8 0.96407 0.96485 0.96562 0.96638 0.96712 0.96784 0.96856 0.96926 0.96995 0.97062
1.9 0.97128 0.97193 0.97257 0.97320 0.97381 0.97441 0.97500 0.97558 0.97615 0.97670
2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.98030 0.98077 0.98124 0.98169
2.1 0.98214 0.98257 0.98300 0.98341 0.98382 0.98422 0.98461 0.98500 0.98537 0.98574
2.2 0.98610 0.98645 0.98679 0.98713 0.98745 0.98778 0.98809 0.98840 0.98870 0.98899
2.3 0.98928 0.98956 0.98983 0.99010 0.99036 0.99061 0.99086 0.99111 0.99134 0.99158
2.4 0.99180 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.99343 0.99361
2.5 0.99379 0.99396 0.99413 0.99430 0.99446 0.99461 0.99477 0.99492 0.99506 0.99520
2.6 0.99534 0.99547 0.99560 0.99573 0.99585 0.99598 0.99609 0.99621 0.99632 0.99643
2.7 0.99653 0.99664 0.99674 0.99683 0.99693 0.99702 0.99711 0.99720 0.99728 0.99736
2.8 0.99744 0.99752 0.99760 0.99767 0.99774 0.99781 0.99788 0.99795 0.99801 0.99807
2.9 0.99813 0.99819 0.99825 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861
3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99900
3.1 0.99903 0.99906 0.99910 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929
3.2 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99948 0.99950
3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99965
3.4 0.99966 0.99968 0.99969 0.99970 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976
3.5 0.99977 0.99978 0.99978 0.99979 0.99980 0.99981 0.99981 0.99982 0.99983 0.99983
3.6 0.99984 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989

1
Table 2 : Standard Normal Table (Z-scores) x→ Zx
P 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
0 infini 3.0902 2.8782 2.7478 2.6521 2.5758 2.5121 2.4573 2.4089 2.3656 2.3263 0.99
0.01 2.3263 2.2904 2.2571 2.2262 2.1973 2.1701 2.1444 2.1201 2.0969 2.0748 2.0537 0.98
0.02 2.0537 2.0335 2.0141 1.9954 1.9774 1.9600 1.9431 1.9268 1.9110 1.8957 1.8808 0.97
0.03 1.8808 1.8663 1.8522 1.8384 1.8250 1.8119 1.7991 1.7866 1.7744 1.7624 1.7507 0.96
0.04 1.7507 1.7392 1.7279 1.7169 1.7060 1.6954 1.6849 1.6747 1.6646 1.6546 1.6449 0.95
0.05 1.6449 1.6352 1.6258 1.6164 1.6072 1.5982 1.5893 1.5805 1.5718 1.5632 1.5548 0.94
0.06 1.5548 1.5464 1.5382 1.5301 1.5220 1.5141 1.5063 1.4985 1.4909 1.4833 1.4758 0.93
0.07 1.4758 1.4684 1.4611 1.4538 1.4466 1.4395 1.4325 1.4255 1.4187 1.4118 1.4051 0.92
0.08 1.4051 1.3984 1.3917 1.3852 1.3787 1.3722 1.3658 1.3595 1.3532 1.3469 1.3408 0.91
0.09 1.3408 1.3346 1.3285 1.3225 1.3165 1.3106 1.3047 1.2988 1.2930 1.2873 1.2816 0.90
0.10 1.2816 1.2759 1.2702 1.2646 1.2591 1.2536 1.2481 1.2426 1.2372 1.2319 1.2265 0.89
0.11 1.2265 1.2212 1.2160 1.2107 1.2055 1.2004 1.1952 1.1901 1.1850 1.1800 1.1750 0.88
0.12 1.1750 1.1700 1.1650 1.1601 1.1552 1.1503 1.1455 1.1407 1.1359 1.1311 1.1264 0.87
0.13 1.1264 1.1217 1.1170 1.1123 1.1077 1.1031 1.0985 1.0939 1.0893 1.0848 1.0803 0.86
0.14 1.0803 1.0758 1.0714 1.0669 1.0625 1.0581 1.0537 1.0494 1.0451 1.0407 1.0364 0.85
0.15 1.0364 1.0322 1.0279 1.0237 1.0194 1.0152 1.0110 1.0069 1.0027 0.9986 0.9945 0.84
0.16 0.9945 0.9904 0.9863 0.9822 0.9782 0.9741 0.9701 0.9661 0.9621 0.9581 0.9542 0.83
0.17 0.9542 0.9502 0.9463 0.9424 0.9385 0.9346 0.9307 0.9269 0.9230 0.9192 0.9154 0.82
0.18 0.9154 0.9116 0.9078 0.9040 0.9002 0.8965 0.8927 0.8890 0.8853 0.8816 0.8779 0.81
0.19 0.8779 0.8742 0.8706 0.8669 0.8632 0.8596 0.8560 0.8524 0.8488 0.8452 0.8416 0.80
0.20 0.8416 0.8381 0.8345 0.8310 0.8274 0.8239 0.8204 0.8169 0.8134 0.8099 0.8064 0.79
0.21 0.8064 0.8030 0.7995 0.7961 0.7926 0.7892 0.7858 0.7824 0.7790 0.7756 0.7722 0.78
0.22 0.7722 0.7688 0.7655 0.7621 0.7588 0.7554 0.7521 0.7488 0.7454 0.7421 0.7388 0.77
0.23 0.7388 0.7356 0.7323 0.7290 0.7257 0.7225 0.7192 0.7160 0.7128 0.7095 0.7063 0.76
0.24 0.7063 0.7031 0.6999 0.6967 0.6935 0.6903 0.6871 0.6840 0.6808 0.6776 0.6745 0.75
0.25 0.6745 0.6713 0.6682 0.6651 0.6620 0.6588 0.6557 0.6526 0.6495 0.6464 0.6433 0.74
0.26 0.6433 0.6403 0.6372 0.6341 0.6311 0.6280 0.6250 0.6219 0.6189 0.6158 0.6128 0.73
0.27 0.6128 0.6098 0.6068 0.6038 0.6008 0.5978 0.5948 0.5918 0.5888 0.5858 0.5828 0.72
0.28 0.5828 0.5799 0.5769 0.5740 0.5710 0.5681 0.5651 0.5622 0.5592 0.5563 0.5534 0.71
0.29 0.5534 0.5505 0.5476 0.5446 0.5417 0.5388 0.5359 0.5330 0.5302 0.5273 0.5244 0.70
0.30 0.5244 0.5215 0.5187 0.5158 0.5129 0.5101 0.5072 0.5044 0.5015 0.4987 0.4958 0.69
0.31 0.4958 0.4930 0.4902 0.4874 0.4845 0.4817 0.4789 0.4761 0.4733 0.4705 0.4677 0.68
0.32 0.4677 0.4649 0.4621 0.4593 0.4565 0.4538 0.4510 0.4482 0.4454 0.4427 0.4399 0.67
0.33 0.4399 0.4372 0.4344 0.4316 0.4289 0.4261 0.4234 0.4207 0.4179 0.4152 0.4125 0.66
0.34 0.4125 0.4097 0.4070 0.4043 0.4016 0.3989 0.3961 0.3934 0.3907 0.3880 0.3853 0.65
0.35 0.3853 0.3826 0.3799 0.3772 0.3745 0.3719 0.3692 0.3665 0.3638 0.3611 0.3585 0.64
0.36 0.3585 0.3558 0.3531 0.3505 0.3478 0.3451 0.3425 0.3398 0.3372 0.3345 0.3319 0.63
0.37 0.3319 0.3292 0.3266 0.3239 0.3213 0.3186 0.3160 0.3134 0.3107 0.3081 0.3055 0.62
0.38 0.3055 0.3029 0.3002 0.2976 0.2950 0.2924 0.2898 0.2871 0.2845 0.2819 0.2793 0.61
0.39 0.2793 0.2767 0.2741 0.2715 0.2689 0.2663 0.2637 0.2611 0.2585 0.2559 0.2533 0.60
0.40 0.2533 0.2508 0.2482 0.2456 0.2430 0.2404 0.2378 0.2353 0.2327 0.2301 0.2275 0.59
0.41 0.2275 0.2250 0.2224 0.2198 0.2173 0.2147 0.2121 0.2096 0.2070 0.2045 0.2019 0.58
0.42 0.2019 0.1993 0.1968 0.1942 0.1917 0.1891 0.1866 0.1840 0.1815 0.1789 0.1764 0.57
0.43 0.1764 0.1738 0.1713 0.1687 0.1662 0.1637 0.1611 0.1586 0.1560 0.1535 0.1510 0.56
0.44 0.1510 0.1484 0.1459 0.1434 0.1408 0.1383 0.1358 0.1332 0.1307 0.1282 0.1257 0.55
0.45 0.1257 0.1231 0.1206 0.1181 0.1156 0.1130 0.1105 0.1080 0.1055 0.1030 0.1004 0.54
0.46 0.1004 0.0979 0.0954 0.0929 0.0904 0.0878 0.0853 0.0828 0.0803 0.0778 0.0753 0.53
0.47 0.0753 0.0728 0.0702 0.0677 0.0652 0.0627 0.0602 0.0577 0.0552 0.0527 0.0502 0.52
0.48 0.0502 0.0476 0.0451 0.0426 0.0401 0.0376 0.0351 0.0326 0.0301 0.0276 0.0251 0.51
0.49 0.0251 0.0226 0.0201 0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 0.50

2
 Table 3 : Table 3 : t-Distribution

Level of
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 0.99 0.995 0.999
Confidence
One-tailed 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.025 0.01 0.005 0.0025 0.0005
Two-tailed 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.02 0.01 0.005 0.001
1 0.1584 0.3249 0.5095 0.7265 1 1.3764 1.9626 3.0777 6.3137 12.706 31.821 63.656 127.32 636.58
2 0.1421 0.2887 0.4447 0.6172 0.8165 1.0607 1.3862 1.8856 2.92 4.3027 6.9645 9.925 14.089 31.6
3 0.1366 0.2767 0.4242 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8408 7.4532 12.924
4 0.1338 0.2707 0.4142 0.5686 0.7407 0.941 1.1896 1.5332 2.1318 2.7765 3.7469 4.6041 5.5975 8.6101
5 0.1322 0.2672 0.4082 0.5594 0.7267 0.9195 1.1558 1.4759 2.015 2.5706 3.3649 4.0321 4.7733 6.8685
6 0.1311 0.2648 0.4043 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 4.3168 5.9587
7 0.1303 0.2632 0.4015 0.5491 0.7111 0.896 1.1192 1.4149 1.8946 2.3646 2.9979 3.4995 4.0294 5.4081
8 0.1297 0.2619 0.3995 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.306 2.8965 3.3554 3.8325 5.0414
9 0.1293 0.261 0.3979 0.5435 0.7027 0.8834 1.0997 1.383 1.8331 2.2622 2.8214 3.2498 3.6896 4.7809
10 0.1289 0.2602 0.3966 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 3.5814 4.5868
11 0.1286 0.2596 0.3956 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.201 2.7181 3.1058 3.4966 4.4369
12 0.1283 0.259 0.3947 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.681 3.0545 3.4284 4.3178
13 0.1281 0.2586 0.394 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 3.3725 4.2209
14 0.128 0.2582 0.3933 0.5366 0.6924 0.8681 1.0763 1.345 1.7613 2.1448 2.6245 2.9768 3.3257 4.1403
15 0.1278 0.2579 0.3928 0.5357 0.6912 0.8662 1.0735 1.3406 1.7531 2.1315 2.6025 2.9467 3.286 4.0728
16 0.1277 0.2576 0.3923 0.535 0.6901 0.8647 1.0711 1.3368 1.7459 2.1199 2.5835 2.9208 3.252 4.0149
17 0.1276 0.2573 0.3919 0.5344 0.6892 0.8633 1.069 1.3334 1.7396 2.1098 2.5669 2.8982 3.2224 3.9651
18 0.1274 0.2571 0.3915 0.5338 0.6884 0.862 1.0672 1.3304 1.7341 2.1009 2.5524 2.8784 3.1966 3.9217
19 0.1274 0.2569 0.3912 0.5333 0.6876 0.861 1.0655 1.3277 1.7291 2.093 2.5395 2.8609 3.1737 3.8833
20 0.1273 0.2567 0.3909 0.5329 0.687 0.86 1.064 1.3253 1.7247 2.086 2.528 2.8453 3.1534 3.8496
21 0.1272 0.2566 0.3906 0.5325 0.6864 0.8591 1.0627 1.3232 1.7207 2.0796 2.5176 2.8314 3.1352 3.8193
22 0.1271 0.2564 0.3904 0.5321 0.6858 0.8583 1.0614 1.3212 1.7171 2.0739 2.5083 2.8188 3.1188 3.7922
23 0.1271 0.2563 0.3902 0.5317 0.6853 0.8575 1.0603 1.3195 1.7139 2.0687 2.4999 2.8073 3.104 3.7676
24 0.127 0.2562 0.39 0.5314 0.6848 0.8569 1.0593 1.3178 1.7109 2.0639 2.4922 2.797 3.0905 3.7454
25 0.1269 0.2561 0.3898 0.5312 0.6844 0.8562 1.0584 1.3163 1.7081 2.0595 2.4851 2.7874 3.0782 3.7251
26 0.1269 0.256 0.3896 0.5309 0.684 0.8557 1.0575 1.315 1.7056 2.0555 2.4786 2.7787 3.0669 3.7067
27 0.1268 0.2559 0.3894 0.5306 0.6837 0.8551 1.0567 1.3137 1.7033 2.0518 2.4727 2.7707 3.0565 3.6895
28 0.1268 0.2558 0.3893 0.5304 0.6834 0.8546 1.056 1.3125 1.7011 2.0484 2.4671 2.7633 3.047 3.6739
29 0.1268 0.2557 0.3892 0.5302 0.683 0.8542 1.0553 1.3114 1.6991 2.0452 2.462 2.7564 3.038 3.6595
30 0.1267 0.2556 0.389 0.53 0.6828 0.8538 1.0547 1.3104 1.6973 2.0423 2.4573 2.75 3.0298 3.646
31 0.1267 0.2555 0.3889 0.5298 0.6825 0.8534 1.0541 1.3095 1.6955 2.0395 2.4528 2.744 3.0221 3.6335
32 0.1267 0.2555 0.3888 0.5297 0.6822 0.853 1.0535 1.3086 1.6939 2.0369 2.4487 2.7385 3.0149 3.6218
33 0.1266 0.2554 0.3887 0.5295 0.682 0.8526 1.053 1.3077 1.6924 2.0345 2.4448 2.7333 3.0082 3.6109
34 0.1266 0.2553 0.3886 0.5294 0.6818 0.8523 1.0525 1.307 1.6909 2.0322 2.4411 2.7284 3.002 3.6007
35 0.1266 0.2553 0.3885 0.5292 0.6816 0.852 1.052 1.3062 1.6896 2.0301 2.4377 2.7238 2.9961 3.5911
36 0.1266 0.2552 0.3884 0.5291 0.6814 0.8517 1.0516 1.3055 1.6883 2.0281 2.4345 2.7195 2.9905 3.5821
37 0.1265 0.2552 0.3883 0.5289 0.6812 0.8514 1.0512 1.3049 1.6871 2.0262 2.4314 2.7154 2.9853 3.5737
Infini
0.1257 0.2533 0.3853 0.5244 0.6744 0.8416 1.0364 1.2816 1.6449 1.96 2.3264 2.5759 2.8072 3.2908
Nor

3
Table 4
χ2-Distribution with n degrees of freedom

0.99 0.98 0.95 0.9 0.85 0.8 0.75 0.7 0.6 0.5 0.4 0.3 0.25 0.2 0.15 0.1 0.05 0.02 0.01 0.001
1 0.000 0.001 0.004 0.016 0.036 0.064 0.102 0.148 0.275 0.455 0.708 1.074 1.323 1.642 2.072 2.706 3.841 5.412 6.635 10.827
2 0.020 0.040 0.103 0.211 0.325 0.446 0.575 0.713 1.022 1.386 1.833 2.408 2.773 3.219 3.794 4.605 5.991 7.824 9.210 13.815
3 0.115 0.185 0.352 0.584 0.798 1.005 1.213 1.424 1.869 2.366 2.946 3.665 4.108 4.642 5.317 6.251 7.815 9.837 11.345 16.266
4 0.297 0.429 0.711 1.064 1.366 1.649 1.923 2.195 2.753 3.357 4.045 4.878 5.385 5.989 6.745 7.779 9.488 11.66 13.277 18.466
5 0.554 0.752 1.145 1.610 1.994 2.343 2.675 3.000 3.656 4.351 5.132 6.064 6.626 7.289 8.115 9.236 11.070 13.388 15.086 20.515
6 0.872 1.134 1.635 2.204 2.661 3.070 3.455 3.828 4.570 5.348 6.211 7.231 7.841 8.558 9.446 10.645 12.592 15.033 16.812 22.457
7 1.239 1.564 2.167 2.833 3.358 3.822 4.255 4.671 5.493 6.346 7.283 8.383 9.037 9.803 10.748 12.017 14.067 16.622 18.475 24.321
8 1.647 2.032 2.733 3.490 4.078 4.594 5.071 5.527 6.423 7.344 8.351 9.524 10.219 11.030 12.027 13.362 15.507 18.168 20.090 26.124
9 2.088 2.532 3.325 4.168 4.817 5.380 5.899 6.393 7.357 8.343 9.414 10.656 11.389 12.242 13.288 14.684 16.919 19.679 21.666 27.877
10 2.558 3.059 3.940 4.865 5.570 6.179 6.737 7.267 8.295 9.342 10.473 11.781 12.549 13.442 14.534 15.987 18.307 21.161 23.209 29.588
11 3.053 3.609 4.575 5.578 6.336 6.989 7.584 8.148 9.237 10.341 11.530 12.899 13.701 14.631 15.767 17.275 19.675 22.618 24.725 31.264
12 3.571 4.178 5.226 6.304 7.114 7.807 8.438 9.034 10.182 11.340 12.584 14.011 14.845 15.812 16.989 18.549 21.026 24.054 26.217 32.909
13 4.107 4.765 5.892 7.041 7.901 8.634 9.299 9.926 11.129 12.340 13.636 15.119 15.984 16.985 18.202 19.812 22.362 25.471 27.688 34.527
14 4.660 5.368 6.571 7.790 8.696 9.467 10.165 10.821 12.078 13.339 14.685 16.222 17.117 18.151 19.406 21.064 23.685 26.873 29.141 36.124
15 5.229 5.985 7.261 8.547 9.499 10.307 11.037 11.721 13.030 14.339 15.733 17.322 18.245 19.311 20.603 22.307 24.996 28.259 30.578 37.698
16 5.812 6.614 7.962 9.312 10.309 11.152 11.912 12.624 13.983 15.338 16.780 18.418 19.369 20.465 21.793 23.542 26.296 29.633 32.000 39.252
17 6.408 7.255 8.672 10.085 11.125 12.002 12.792 13.531 14.937 16.338 17.824 19.511 20.489 21.615 22.977 24.769 27.587 30.995 33.409 40.791
18 7.015 7.906 9.390 10.865 11.946 12.857 13.675 14.440 15.893 17.338 18.868 20.601 21.605 22.760 24.155 25.989 28.869 32.346 34.805 42.312
19 7.633 8.567 10.117 11.651 12.773 13.716 14.562 15.352 16.850 18.338 19.910 21.689 22.718 23.900 25.329 27.204 30.144 33.687 36.191 43.819
20 8.260 9.237 10.851 12.443 13.604 14.578 15.452 16.266 17.809 19.337 20.951 22.775 23.828 25.038 26.498 28.412 31.410 35.020 37.566 45.314
21 8.897 9.915 11.591 13.240 14.439 15.445 16.344 17.182 18.768 20.337 21.992 23.858 24.935 26.171 27.662 29.615 32.671 36.343 38.932 46.796
22 9.542 10.600 12.338 14.041 15.279 16.314 17.240 18.101 19.729 21.337 23.031 24.939 26.039 27.301 28.822 30.813 33.924 37.659 40.289 48.268
23 10.196 11.293 13.091 14.848 16.122 17.187 18.137 19.021 20.690 22.337 24.069 26.018 27.141 28.429 29.979 32.007 35.172 38.968 41.638 49.728
24 10.856 11.992 13.848 15.659 16.969 18.062 19.037 19.943 21.652 23.337 25.106 27.096 28.241 29.553 31.132 33.196 36.415 40.270 42.980 51.179
25 11.524 12.697 14.611 16.473 17.818 18.940 19.939 20.867 22.616 24.337 26.143 28.172 29.339 30.675 32.282 34.382 37.652 41.566 44.314 52.619
26 12.198 13.409 15.379 17.292 18.671 19.820 20.843 21.792 23.579 25.336 27.179 29.246 30.435 31.795 33.429 35.563 38.885 42.856 45.642 54.051
27 12.878 14.125 16.151 18.114 19.527 20.703 21.749 22.719 24.544 26.336 28.214 30.319 31.528 32.912 34.574 36.741 40.113 44.140 46.963 55.475
28 13.565 14.847 16.928 18.939 20.386 21.588 22.657 23.647 25.509 27.336 29.249 31.391 32.620 34.027 35.715 37.916 41.337 45.419 48.278 56.892

4
Tables 5 and 6 : Shewhart Control Charts

μ and σ are unknowns and are knowns

Chart Type
CL UCL & LCL CL UCL & LCL

X A R
X X μ μ Aσ
X A S
R R D R ,D R d σ D σ ,D σ
S S B S, B S , c σ B σ ,B σ

Table 7: Some Main Formulas:

Confidence Intervals for the Mean (Large Samples) !

√ √
2
(n  1)s (n  1)s 2
Confidence Interval for σ: 2 σ 
XR X L2

Test statistic for mean

Test statistic for standard deviation

Test statistic for proportions

Standard deviation estimation:

Sample size for one-sample z test Sample Size for Control Chart